arX

iv:1

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5741

v2 [

cond

-mat

.sta

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ech]

18

Nov

201

0 Dynamical Transition in the Open-boundary Totally

Asymmetric Exclusion Process

A. Proeme, R. A. Blythe and M. R. Evans

SUPA, School of Physics and Astronomy, University of Edinburgh, Mayfield Road,

Edinburgh EH9 3JZ, United Kingdom

E-mail: arno.proeme@ed.ac.uk, r.a.blythe@ed.ac.uk, m.evans@ed.ac.uk

Abstract. We revisit the totally asymmetric simple exclusion process with open

boundaries (TASEP), focussing on the recent discovery by de Gier and Essler that the

model has a dynamical transition along a nontrivial line in the phase diagram. This

line coincides neither with any change in the steady-state properties of the TASEP,

nor the corresponding line predicted by domain wall theory. We provide numerical

evidence that the TASEP indeed has a dynamical transition along the de Gier–Essler

line, finding that the most convincing evidence was obtained from Density Matrix

Renormalisation Group (DMRG) calculations. By contrast, we find that the dynamical

transition is rather hard to see in direct Monte Carlo simulations of the TASEP. We

furthermore discuss in general terms scenarios that admit a distinction between static

and dynamic phase behaviour.

Date: October 26, 2010

Dynamical Transition in the TASEP 2

1. Introduction

As a rare example of a class of exactly-solvable, nonequilibrium interacting particle

systems, asymmetric simple exclusion processes of various types have found favour with

researchers in statistical mechanics, mathematical physics and probability theory [1–4].

These systems comprise hard-core particles hopping in a preferred direction on a one-

dimensional lattice and have been used to describe systems as diverse as traffic flow [5],

the dynamics of ribosomes [6] and molecular motors [7, 8], the flow of hydrocarbons

through a zeolite pore [9], the growth of a fungal filament [10], and in queueing

theory [11,12]. The physical interest lies mainly in the rich phase behaviour that arises

as a direct consequence of being driven away from equilibrium.

Of particular interest are the open boundary cases. Here the system can be thought

of as being placed between two boundary reservoirs, generally of different densities. The

two reservoirs enforce a steady current across the lattice, and therewith a nonequilibrium

steady state. It was first argued by Krug [13] that as the boundary densities are varied

nonequilibrium phase transitions may occur in which steady state bulk quantities—

such as the mean current or density—exhibit nonanalyticities. Such phase transitions

were seen explicitly in first a mean-field approximation [6, 14] and then in the exact

solution [15, 16] of the totally asymmetric exclusion process with open boundaries,

hereafter abbreviated as TASEP, and of related processes [4]. Since then arguments

have been developed to predict the phase diagram of more general one-dimensional

driven diffusive systems [17–19].

Our interest in this work is in the distinction between two phase diagrams for

the TASEP: one of which characterises the steady-state behaviour, and the other the

dynamics. The static phase diagram [15,16] is shown in Fig. 1(a) where the left boundary

density is α and the right is 1− β. There are three possible phases in the steady state:

a low density (LD) phase where the bulk density is controlled by the left boundary and

is equal to α; a high density (HD) phase where the bulk density is controlled by the

right boundary and is equal to 1 − β; a maximal current (MC) phase where the bulk

density is 12. The high and low density phases are further divided into subphases (e.g.

LD1, LDII) according to the functional form of the spatial decay of the density profile

to the bulk value near the non-controlling boundary. In these subphases the lengthscale

over which the decay is observable remains finite as the system size is increased. This

change at the subphase boundaries in the form of the decay is thus not a true phase

transition in the thermodynamic sense.

More recently, de Gier and Essler have performed an exact analysis of the ASEP’s

dynamics [20–22] and derived the the longest relaxation times of the system by

calculating the gap in the spectrum using the Bethe ansatz. The analysis builds on

directly related work on the XXZ spin chain with nondiagonal boundary fields [23, 24].

The dynamical phase diagram they obtain, in which phases are associated with different

functional forms of the longest relaxation time, is illustrated in figure 1(b). The same

phases (high density, low density and maximal current) are found as in the static phase

Dynamical Transition in the TASEP 3

0 0.5 10

0.5

1

PSfrag replacements j = α(1− α)

j = α(1− α)

ρ = α

ρ = α

ρ = 1− βρ = 1− β

j = β(1− β)j = β(1− β)LDI

LDII

HDIIHDI

MC

ρ = 1

2

j = 1

4

α

β

(a)

0 0.5 10

0.5

1

PSfrag replacements

LDI′

LDII′

HDII′

HDI′

MC

α

β

(b)

Figure 1. (a) Static and (b) dynamic phase diagrams for the TASEP. Solid lines

indicate thermodynamic phase transitions at which the current and bulk density are

nonanalytic. Dotted lines indicate subphase boundaries. In the static case, the density

profile a finite distance from the boundary is nonanalytic. In the dynamic case, the

longest relaxation time exhibits a nonanalyticity.

diagram, however, a hitherto unexpected dynamical transition line which subdivides the

low density and high density phases is now apparent. This line which we shall refer to

as the de Gier–Essler (dGE) line replaces the subphase boundaries at α = 1/2, β < 1/2

β = 1/2, α < 1/2 in the static phase diagram and, for example, subdivides the low

density region into LDI′, LDII′.

Although there is nothing to rule out the prediction of dGE of a dynamical

transition at a distinct location to any static transition, the result came as something of

a surprise. This is because the static phase diagram had been successfully interpreted in

terms of an effective, dynamical theory thought to be relevant for late-time dynamics,

referred to as domain wall theory (DWT). We shall review DWT more fully in

Section 2.4. For the moment we note that DWT correctly predicts the static subphase

boundary and the associated change in the decay of the density profile, but does not

predict the dGE line. Therefore, it was previously thought that a dynamical transition

also occurred at the subphase boundary and initial calculations of relaxation times

appeared to be consistent with this [25, 26].

Our primary goal in this work is to establish numerically that a dynamical transition

does indeed occur along the dGE line rather than at the static subphase boundary. We

used two distinct approaches. First, we carried out direct Monte Carlo simulations of

the TASEP dynamics and identified the dominant transient in three time-dependent

observables. We find that these simulations are consistent with a dynamical transition

at the dGE line but are not sufficiently accurate to rule out the scenario of the dynamical

transition occurring at the subphase boundary.

Dynamical Transition in the TASEP 4

We instead turn to a Density Matrix Renormalisation Group (DMRG) approach [27]

to acquire convincing evidence that a dynamical transition occurs at the dGE line. The

DMRG is an approximate means to obtain the lowest-lying energies and eigenstates of

model Hamiltonians with short-range interactions. Although originally developed in the

context of the Hubbard and Heisenberg models [28] (see also [29] for a comprehensive

review), it has also been applied to nonequilibrium processes such as reaction-diffusion

models [30]. In particular, it has also been used to investigate the spectrum of the

TASEP [26]. However, the dynamical transition was not evident from the data presented

in that work—perhaps because it had not been predicted at that time. By revisiting

this approach, we obtain estimates of the longest relaxation time that allow us to rule

out the domain wall theory prediction for the dynamical transition.

The paper is organised as follows. We begin in Section 2 by recalling the definition

of the TASEP, and by reviewing in more detail the static and dynamic phase diagrams

discussed above. Then, in Section 3, we present our Monte Carlo simulation data,

followed by the DMRG results in Section 4. In Section 5 we return to more general

questions about the distinction between the static and dynamic phase diagram with

particular reference to different theoretical approaches to the TASEP. We conclude in

Section 6 with some open questions.

2. Model definition and phase diagrams

Although we have alluded to the basic properties of the totally asymmetric simple

exclusion process (TASEP) in the introduction, in the interest of a self-contained

presentation we provide in this section a precise model definition and full details of

the static and dynamic phase diagrams.

2.1. Model definition

The TASEP describes the biased diffusion of particles on a one-dimensional lattice with

L sites. No more than one particle can occupy a given site, and overtaking is prohibited.

The stochastic dynamics are expressed in terms of transition rates, for example a particle

on a site i in the bulk hops to the right as a Poisson process with rate 1, but only if site

i+ 1 is unoccupied: At the left boundary only particle influx takes place, with rate α,

and at the right boundary only particle outflux takes place, with rate β, as shown in

Fig. 2. The corresponding reservoir densities are α at the left and 1− β at the right.

2.2. Static phase diagram

As noted in the introduction, the TASEP static phase diagram can be determined

from, for example, exact expressions for the current and density profile in the steady

state [15,16]. The current takes three functional forms according to whether it is limited

by a slow insertion rate (LD: α < β, α < 12), by a slow exit rate (HD: β < α, β < 1

2)

or by the exclusion interaction in the bulk (MC: α > 12, β > 1

2). This behaviour of

Dynamical Transition in the TASEP 5

Figure 2. The dynamics of the totally asymmetric simple exclusion process (TASEP)

with open boundaries. Arrows show moves that may take place, and the labels indicate

the corresponding stochastic rates.

the current leads to the identification of the thermodynamic phases separated by solid

lines in Fig. 1(a). Along these lines there are nonanalyticities in both the current and

the density far from either boundary. The relevant expressions for these quantities are

shown on Fig. 1(a).

The density near one of the boundaries is nonanalytic along the lines α = 12and

β = 12in the HD and LD phases respectively. These subphases are shown dotted in

Fig. 1(a). Inspection of the functional form of the density profile reveals that these

are not true thermodynamic phase transitions, in the sense that the deviation from

the bulk value extends only a finite distance into the bulk, and thus contributes only

subextensively to the nonequilibrium analogue of a free energy (see e.g., [31] for the

definition of such a quantity). To be more explicit, consider for example the behaviour

near the right boundary in the low-density phase. Here the bulk density is ρ = α.

The mean occupancy of the lattice site positioned a distance j from the right boundary

approaches in the thermodynamic limit L→ ∞ the form [4, 15, 16]

ρL−j ∼α+ cI(β)

(α(1−α)β(1−β)

)jα < β < 1

2(LDI)

α+ cII(α, β)[4α(1−α)]j

j3/212< β (LDII)

. (1)

In these expressions, cI and cII are functions of the boundary parameters that we leave

unspecified here so as to focus on the lengthscale of the exponential decay from the

right boundary. As β is increased from zero, the decay length increases until β = 12.

Then the decay length is constant, and the exponential is modulated by a power law.

The density profile at the left boundary exhibits the same kind of nonanalyticity in the

high-density phase as α is increased through 12as a consequence of the particle-hole

symmetry, ρi−1(α, β) = 1− ρL−i(β, α), exhibited by the model.

2.3. Dynamical phase diagram

The dynamic phase diagram is obtained by examining a quantity referred to as the

gap by de Gier and Essler [20–22]. This is simply the largest nonzero eigenvalue of

the transition matrix governing the continuous-time stochastic dynamics of the TASEP.

More formally, one starts with the master equation

d

dt|P (t)〉 =M |P (t)〉 , (2)

where the matrixM encodes the transition rates and |P (t)〉 is the vector of probabilitiesfor each configuration. Because M is a stochastic matrix it is guaranteed [32] by the

Dynamical Transition in the TASEP 6

Perron-Frobenius theorem to have eigenvalues satisfying

λ0 = 0 > Re(λ1) ≥ Re(λ2) ≥ . . . . (3)

The spectrum corresponds to a set of relaxation times τi = −(Reλi)−1. The longest

(non-infinite) relaxation time is τ1, and the associated eigenvalue λ1 is the gap which we

will henceforth denote by the symbol ε. At long times the relaxation of any observable

is expected to decay exponentially with a characteristic timescale τ1, a fact we will later

use to estimate the gap from Monte Carlo simulations.

The thermodynamic phase boundaries are found by identifying lines along which

the gap vanishes, indicating the coexistence of two stationary eigenstates (phases). The

exact calculations of de Gier and Essler [20, 21] show that, in the thermodynamic limit

L → ∞, the gap vanishes along the line α = β < 12that separates the HD and LD

phases. The gap also vanishes in the entirety of the MC phase, reflecting the generic

long-range (power-law) correlations seen in this phase [15]. Thus at this level, the static

and dynamic phase diagrams coincide.

Where they differ is in the subdivision of the high- and low-density phases‡, inwhich the gap remains finite in the limit L → ∞. There is a region, marked LDI′ and

HDI′ on Fig. 1(b), within which the gap assumes the asymptotic expression

ε(L) = −α− β +2

(ab)1

2 + 1− π2

(ab)1

2 − (ab)−1

2

L−2 +O(L−3) (4)

in which

a =1− α

αand b =

1− β

β. (5)

Within the low-density phase (α < β, α < 12), this form of the gap applies for values of

β < βc where

βc(α) =

[1 +

(α

1− α

) 1

3

]−1

. (6)

Likewise, in the high-density phase (β < α, β < 12), the region within which the gap is

given by (4) is bounded by α < αc where

αc(β) =

1 +

(β

1− β

) 1

3

−1

. (7)

In the remainder of the low-density phase, α < 12, β > βc the gap takes the form

ε(L) = −α− βc +2

(abc)1

2 + 1− 4π2

(abc)1

2 − (abc)− 1

2

L−2 +O(L−3) . (8)

Finally, we have by symmetry that when β < 12, α > αc,

ε(L) = −αc − β +2

(acb)1

2 + 1− 4π2

(acb)1

2 − (acb)− 1

2

L−2 +O(L−3) . (9)

‡ de Gier and Essler [20–22] refer to these as “massive” phases by analogy with the quantum spin

chains; we shall only use the terminology associated with the static phase diagram to avoid confusion.

Dynamical Transition in the TASEP 7

Figure 3. Exact L → ∞ gap (black, dashed) as a function of β for α = 0.3, described

by the HDI′ equation (4) (red) for β < βc (βc ≈ 0.57, indicated), and by equation (8)

(green) for β > βc.

In these expressions,

ac =1− αc

αc

and bc =1− βcβc

. (10)

The boundaries between the dynamic subphases—the de Gier–Essler (dGE) lines—are

shown dotted in Fig. 1(b). Note that the coefficient of the L−2 term in this asymptotic

expansion is discontinuous across the dynamical transition line.

In order to illustrate the behaviour of the gap along the static and dynamic

transition lines, we plot it as a function of β at some α < 12in Fig. 3. The most

striking feature is the constancy of the gap above the nontrivial critical point βc. Later

in this work, we will use the constancy of the gap above a critical threshold as an

empirical means to identify the dynamical transition point.

2.4. Domain wall theory

An alternative approach to study the TASEP dynamics is domain wall theory

(DWT) [33]. Although much simpler than the Bethe ansatz approach of de Gier and

Essler, it correctly predicts the static phase boundaries and the analytical form of the

gap in a region of the phase diagram. However, it predicts dynamic subphases that

are distinct from those found by de Gier and Essler, and that correspond to the static

subphases. In the numerical study that follows, it will be important to be able to

distinguish between these two sets of predictions, and so we summarise DWT here.

The basis of this approach is to assume that collective relaxational dynamics are

effectively reducible to a single coordinate describing the position of an interface, or

domain wall. The wall separates a domain of density ρ− and current j− to the left from

a domain of density ρ+ and current j+ = ρ+(1− ρ+) to the right. Each domain is taken

to possess the steady state characteristics imposed by the boundary reservoir on that

Dynamical Transition in the TASEP 8

0 0.5 10

0.5

1

PSfrag replacements

LDI

LDII

HDIIHDI

MC

j = 1

4

α

β

ρ+ = 1

2

ρ− = α

ρ+ = 1− β

ρ− = α

ρ+ = 1− β

ρ− = α

ρ+ = 1− β

ρ− = 1

2

Figure 4. The phase diagram obtained from domain wall theory. ρ+ and ρ− indicate

the densities of the left and right domains. In the domain wall theory, static and

dynamic transitions coincide along the dashed lines.

particular side of the wall, with ρ+ and ρ− therefore as in Fig. 4. The effective theory

is expected to be exact along the line α = β < 1/2 where the exact properties of the

system such as density profile [15, 16] and current fluctuations [34] are recovered.

Let us consider the case α < 1/2 and β < 1/2. The motion of the wall is then

described microscopically as a random walker with left and right hopping rates D− and

D+ given respectively by imposing mass conservation on the fluxes into and out of the

wall [25]:

D− =j−

ρ+ − ρ−=

α(1− α)

1− α− βD+ =

j+

ρ+ − ρ−=

β(1− β)

1− α− β, (11)

For α > β the random walk is biased to the left and in the stationary state the domain

wall is localised at the left boundary and the bulk density is given by ρ+ = 1− β. For

α < β the random walk is biased to the right and in the stationary state the domain

wall is localised at the right boundary and the bulk density is given by ρ− = α. Thus

the first order transition at α = β is correctly predicted.

Moreover, the gap in the resulting spectrum for large L given [25] is given by

εDWT = −D+ −D− + 2√D+D−

(1− π2

2L2+O(L−3)

). (12)

Remarkably this expression is identical to (4) to order 1/L2. Thus in the region

α < 1/2, β < 1/2 DWT correctly predicts the gap.

When β > 1/2 so that ρ+ = 1/2, one can take j+ in (11) to depend on the size ℓ

of the right-hand domain. That is, one puts j+ → j+(ℓ) equal to the stationary current

in a TASEP of size ℓ in the maximal current phase. This implies the large ℓ behaviour

j+ ≃ 1

4(1 +

3

2ℓ) (13)

Dynamical Transition in the TASEP 9

Figure 5. Comparison of the gap for L → ∞ from domain wall theory (dashed line)

with the exact gap (solid line) at α = 0.1 (for which βc ≈ 0.675).

This results in a modification of the density profile to an exponential spatial decay

modulated by a power law with power 3/2, similar to Eq. 1 (LDII).

In brief, the DWT is remarkably successful, correctly predicting the static phase

diagram (including subphases), and the exact thermodynamic gap function found by

de Gier and Essler in the region α < 1/2 and β < 1/2. It differs in that the dynamic

subphases are not given by the dGE lines, but the static subphase transition lines. Thus,

the region within which the gap is constant is different in the two theories, as indicated

by Fig. 5.

3. Evidence for the dynamical transition from Monte Carlo simulations

We now describe attempts to measure the gap in Monte Carlo simulations of the TASEP,

with a particular interest in distinguishing between the dGE and DWT predictions. As

noted previously, the ensemble average of any time-dependent observable O(t) should

be dominated at late times by an exponential decay e−|λ1|t to its stationary value 〈O〉.Recall that λ1 is the largest nonzero eigenvalue of the matrixM appearing in the master

equation (2), and that all nonstationary eigenvalues have negative real part. In principle,

therefore, all one needs to do is pick an observable, and measure its late-time decay rate.

In practice, this is made difficult by the fact that all decay modes may contribute to

〈O(t)〉, and that by the time the higher modes have decayed away, the residual signal

〈O(t)〉 − 〈O〉 may be extremely small and swamped by the noise.

Since we are interested in the time-dependence of an observable, we employ a

continuous-time (Gillespie) algorithm [35] to simulate the TASEP dynamics. More

precisely, we maintain a list of events (i.e., a particle hopping to the next site, or entering

or leaving the system) that can take place given the current lattice configuration.

A particular event i is then chosen with a probability proportional to its rate ωi as

specified in Section 2. A time variable is then incremented by an amount chosen from

Dynamical Transition in the TASEP 10

an exponential distribution with mean (∑

i ωi)−1. In this way, a member of the ensemble

of all continuous-time trajectories of the TASEP dynamics from some prescribed initial

condition is generated with the appropriate probability.

3.1. Decay of total occupancy to stationarity

We consider first the decay of the total number of particles on the lattice, N(t), to its

stationary value as measured by the function

R(t) =〈N(t)〉 − 〈N〉〈N(0)〉 − 〈N〉 . (14)

In this expression, angle brackets denote an average over an ensemble of initial conditions

and over stochastic trajectories. In each simulation run, an initial condition was

constructed in which each site was independently occupied with a probability p = 1−β

(in the LD phase) or p = α (in the HD phase). In the bulk, these densities then relax

to the steady-state values displayed on Fig. 1(a).

Once R(t) has been sampled over multiple trajectories, the task is to identify a

time window over which one can fit an exponential decay and therewith estimate a

gap. The start and end points of this windows are both crucial. If it starts too early,

then one may expect contributions from subdominant transients (i.e., the decays at rate

λ2, λ3, . . .) to systematically skew the estimate of the gap. If it ends too late, noise may

instead dominate the estimate.

The noise at the top end we handle by examining the behaviour of local decay rates

µi =lnR(ti+i)− lnR(ti)

ti+1 − ti(15)

where ti and ti+1 are successive time points at which R(t) was sampled. At late times,

one should find µi+1/µi → 1. Strong deviations from unity indicate the dominance of

noise, and we rejected points after which the magnitude of this ratio exceeded a critical

value. For our data sets, we found that 5 was a suitable choice for this value: see

Fig. 6(a) for an example.

The bottom end of the window was chosen by maximising a goodness-of-fit measure

to a fit of the exponential f(t) = ae−λt to data points within the window. We adopted

the adjusted coefficient of determination [36]

R2 = 1−1

n−k

∑i(R(ti)− f(ti))

2

1n−1

∑i(R(ti)− R)2

. (16)

as our goodness-of-fit measure, varying the start of the window until this quantity was

maximised. In this expression R is the arithmetic mean of R(t) over the set of n times

ti falling within the window, and k is the number of free parameters in the fit function

f(t), i.e., k = 2. This goodness-of-fit measure trades off the increased quality of the

fit obtained by discarding data points against the increasing uncertainty in the fitting

parameters that comes with the noisier data at the top end of the window. We show

in Fig. 6(b) an example of how the goodness-of-fit varies with the size of the window.

Dynamical Transition in the TASEP 11

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0 100 200 300 400-4

-2

0

2

4

6

8

t

Μi+

1�Μ

i

(a)

æ

æ

æ

æ

æ

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ææææ æ

æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ

80 120 160 200

0.99980

0.99985

0.99990

0.99995

1.00000

ts

R2

(b)

Figure 6. Criteria for choosing the range of data over which to fit a single exponential

to the decay of the density to stationarity. (a) The ratio of gradients µi+1/µi which

should be close to unity where a single exponential is a good fit. At large times, this

ratio becomes noisy; the presence of extreme values (here, a magnitude larger than 5)

is used to identify the onset of noise. (b) The behaviour of R2 as a function of the

start of the window. We choose this such that R2 is maximised. In both cases, L = 50,

α = 0.3 and β = 0.64.

We remark that the relative flatness of R2 above the optimal starting time can be

ascribed to the fact that only a single decay mode remains in this region. Although this

procedure is a little cumbersome to apply to each data set, we found it preferable to

fitting multiple exponentials to the data for R(t), which leads to large uncertainties in

the measured gap.

We show results in Fig. 7, in which the gap is plotted as a function of β for fixed

α = 0.3 for a range of system sizes. At the smallest system sizes (L = 10) data

from Monte Carlo simulation are in agreement with results obtained through exact

diagonalisation of the matrix M . As the system size is increased beyond the sizes for

which exact techniques remain tractable, the curves approach those predicted by dGE

and DWT. However, it is not possible to discern from the largest system simulated,

L = 150, which of these theories is more appropriate.

We therefore attempt to extrapolate to the limit L → ∞ by fitting a function of

the form ε+ a2L−2 + a3L

−3 + a4L−4 to estimates of the gap at finite system sizes L but

fixed α and β. This particular fitting function was found to have the best goodness-of-fit

(taking into account the varying number of parameters) out of those that were tried,

and furthermore had the most uniform goodness-of-fit as a function of β. The results

of this extrapolation procedure are shown in Fig. 8. These data appear to exclude the

functional form for the domain wall theory, although the uncertainties in the estimated

gaps are such that further evidence is needed in order to confidently rule it out.

3.2. Decay of other observables

One possible means to reduce the uncertainty in the measured gap is to try different

observables O(t) in the hope that contributions from subdominant decay modes—and

in particular the second longest-lived mode that decays at rate λ2—are reduced and

Dynamical Transition in the TASEP 12

Figure 7. The gap as a function of β at α = 0.3 and system sizes L ranging from

L = 40 to L = 150 (from bottom to top) obtained from Monte Carlo simulations. The

L → ∞ dGE (solid line) and DWT (dashed line) predictions are shown for comparison:

even with the largest system L = 150 simulated, one cannot distinguish between them

numerically.

Figure 8. The gap extrapolated to L = ∞ from the data shown in Fig. 7. Again the

dGE (solid line) and DWT (dashed line) predictions are shown for comparison: here

the data are potentially more compatible with the former than the latter.

thereby allow a more accurate estimate of λ1. We considered two candidates.

The first was the autocorrelation function of the total occupancy, O(t) =

N(t0)N(t0 + t), where t0 is some fixed time point in the steady state. The analogue of

the function R(t) is

χ(t) =〈N(t0 + t)N(t0)〉 − 〈N〉2

〈N2〉 − 〈N〉2 (17)

where here all averages are taken in the steady state. Numerically, this quantity is

Dynamical Transition in the TASEP 13

Figure 9. Interactions between first class (circles) and second class (hexagons)

particles. Second class particles hop can forward into empty sites and exchange places

with first-class particles behind them. Note that in our simulations, at most one second-

class particle was present at any instant, and that the behaviour at the boundaries was

chosen so as not to affect the motion of first-class particles (see text).

slightly more convenient than N(t), since one can obtain 〈N(t0 + t)N(t0)〉 by sampling

at different t0 and t from a single simulation run that has reached the steady state.

The other observable we investigated was the position of a second class particle

[37–39]. Like a first-class particle, a second-class particle hops to the right into vacant

sites. However, it may also exchange places with a particle occupying the site to its left.

Both of these processes take place at unit rate, and first-class hop to the right at the

same rate, irrespective of whether the site in front of them is empty or occupied by a

second-class particle—see Fig. 9. So as not to affect the entry of first-class particles into

the system, a second-class particle on the left-boundary site is forced to exit if a first-

class particle attempts to enter. It is reinserted as soon as the left-boundary site becomes

vacant again. Second-class particles are prevented from exiting at the right boundary.

In these simulations one can measure the position of the second class particle, x(t),

starting from an initial condition where each site is occupied by a first-class particle

with density ρ (as described above) except for the left boundary site, which is always

initially occupied by the second-class particle. Here, the analogue of R(t) is

ξ(t) =〈x(t)〉 − 〈x〉x(0)− 〈x〉 . (18)

Gaps obtained from these two functions χ(t) and ξ(t) are compared with those

obtained from R(t) in Fig. 10 at various system sizes. We see that there are no

systematic differences between these functions at the system sizes studied, and therefore

that they are unlikely to provide improved estimates of the gap in the thermodynamic

limit L→ ∞. We therefore conclude that whilst Monte Carlo simulation data is possibly

more consistent with the de Gier–Essler prediction for the gap than the domain wall

theory, the effect of the changing gap on the relaxation of a typical observable is very

small and hard to disentangle from the noise.

4. Evidence for the dynamical transition from density matrix

renormalisation

Given the difficulties in measuring the gap from Monte Carlo simulations, we now

turn to a fundamentally different numerical approach, namely the density matrix

renormalisation group (DMRG) [27] that was briefly discussed in the introduction.

Dynamical Transition in the TASEP 14

Figure 10. The gap as a function of β as obtained by consideration of the relaxation

of three different observables. Solid lines show data obtained from the relaxation of the

density for system sizes L = 10, 20, 30, 50, 150 (bottom to top). Alongside the lower

three curves are plotted data from the stationary density auto-correlation function (17)

(dashed lines), and alongside the upper two data from the position of a single second-

class particle (18) (large symbols). All three observables yield consistent results.

We begin by recapitulating the essential features of the DMRG approach before

demonstrating that it does indeed show a dynamical transition along the de Gier–Essler

line.

4.1. DMRG algorithm for the TASEP gap

The density matrix renormalisation group (DMRG) procedure builds an approximation

to the transition matrix M that appears in the master equation (2). The basic idea is

to repeatedly add sites to the system and renormalise the enlarged transition matrix

so that its dimensionality remains constant as the system size grows. In this way,

the approximated transition matrix remains sufficiently small that its spectrum (and

in particular the gap) can be determined using standard numerical diagonalisation

methods. The success of this procedure relies on finding a reduced basis set at

each renormalisation step that allows the lowest-lying eigenstates to remain well

approximated as sites are added.

Since the TASEP has open boundaries, the extension of the lattice is achieved in

this instance by adding sites to the middle of the system rather than at one of the

ends [26]. The system is thus divided into two ‘blocks’: the left and right halves of the

lattice. It is extended in each iteration by adding a site at the right-hand end of the

left block, and the left-hand end of the right block. In order to understand the results

that come out of the procedure, it is worth explaining in a little more detail how the

renormalisation is actually achieved—full details are provided in Appendix A.

We outline the procedure in terms of the transformation applied to the left block.

Dynamical Transition in the TASEP 15

Figure 11. Illustration of the DMRG procedure applied to the left block. Before the

transformation, the state of the block is specified by 1 ≤ p ≤ m and σ = 0, 1. After

adding a new site to the end of the block, the original ℓ sites are renormalised so that

they are specified by the two numbers 1 ≤ p ≤ m and σ = 0, 1. The right block is a

mirror image of the left, and treated the same way.

Suppose that at the start of the iteration, the block comprises ℓ lattice sites. The set

of states the block can be in is specified by two ‘quantum’ numbers, p = 1, 2 . . . , m and

σ = 0, 1. The quantity p indexes the configuration of the first ℓ − 1 sites, and σ the

occupancy of the rightmost site. The first step of the renormalisation is to construct

some basis in the 2m-dimensional space spanned by p and σ and, crucially, retain only

m of these basis vectors. This defines a renormalised index p = 1, 2, . . . , m for the

entire ℓ-site block. A new site is then added to the right hand side of the lattice. The

configuration of this additional site is specified by the renormalised coordinate σ. See

Fig. 11. The reason for keeping the rightmost site in the ‘physical’ (site occupancy)

basis is that one needs to project the transition matrix for hops from site ℓ to ℓ+1 onto

the space spanned by p and σ. These transition rates are initially only known in this

physical basis. Thus through this procedure, we obtain a (truncated) representation of

the transition matrix for an ℓ + 1-site system in a space of the same dimensionality as

that used to describe the ℓ-site system.

The same transformation is applied to the right block, the only difference being that

it is the leftmost site that is expressed in the physical basis. Interactions between the two

blocks enter through the construction of the renormalised indices p as we now describe.

First, a transition matrixM ′ for the entire system of 2ℓ sites is constructed by combining

the matrices for the two halves, and by adding an interaction term that allows a particle

in the left block to hop to the right block. Again, having the internal two sites specified

in the physical basis helps here. This transition matrix is then diagonalised. However,

it is not these eigenstates that are used to perform the renormalisation. Rather, a

density matrix, which is a symmetric combination of the two longest-lived eigenstates

of M ′, is constructed, and eigenvectors of this density matrix are instead used for the

renormalisation. The idea is that this form of the density matrix allows the stationary

and longest-lived transient state to be accurately represented in large systems. The

specific form of the density matrix, and the prescription for obtaining the truncated

basis set, is given in Appendix A. For further details about the principles behind DMRG

and other applications we refer to [27, 40, 41], and especially [29].

Dynamical Transition in the TASEP 16

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0 50 100 150 200 250-0.10

-0.08

-0.06

-0.04

-0.02

0.00

L

¶

Figure 12. Stable DMRG data for the gap (red crosses) and the exact thermodynamic

gap (black line) at α = 0.4 and β = 0.75.

4.2. DMRG results

We used the DMRG algorithm outlined above to estimate the gap for a given

combination of α and β by starting with an exact diagonalisation of the L = 8 system,

and keeping m = 16 eigenvectors of the density matrix in each renormalisation step. In

principle, one ought to be able to access arbitrarily large system sizes with this method.

In practice—and as was also noted in [26]—the algorithm eventually goes unstable,

which is typically manifested through the gap acquiring an imaginary part. We simply

ignore data for system sizes where the instability is judged to have kicked in.

Although we can access larger system sizes with the DMRG approach than was

possible with Monte Carlo (e.g., L up to about 250, as shown in Fig. 12), it is still

necessary to extrapolate to the thermodynamic limit, L → ∞. We follow a similar

procedure to that described in Section 3.1. That is, we specify a finite-size fitting

function of the form f(L) = ε+ a2L−2, and adjust the smallest value of L used for the

fit. Again, we use R2 as a goodness-of-fit measure, i.e., Eq. (16) with t replaced by L.

We show in Fig. 13(a) how the goodness of fit varies with the minimum value of L used

in the fit; choosing the optimal (largest) value yields an estimate of the gap that we

observe from Fig. 13(b) that appears more consistent with the de Gier–Essler prediction

than domain wall theory.

We show the DMRG estimate of the gap obtained in this way as a function of β

in Fig. 14. The agreement with the analytical prediction common to dGE and DWT

below β < 12is very good. For larger β the data are scattered around an apparently

constant value, indicating the presence of inaccuracies in the DMRG algorithm and/or

the extrapolation of L → ∞. If we regard these data as independent measurements of

the same value, we can obtain an estimate of the uncertainty in the constant value of

the gap above some critical point by simply calculating the standard deviation. We find

the DMRG gap to approach the constant value ε = −0.00125(9) when α = 0.4. For

Dynamical Transition in the TASEP 17

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0.0000 50 100 150 200 250

Lmin

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(b)

Figure 13. (a) Coefficient of determination R2 and (b) extrapolated gap, as a

function of Lmin for DMRG data corresponding to α = 0.4, β = 0.75. The optimal

choice of Lmin lies just below 200, for which the extrapolated gap very closely matches

the de Gier-Essler result and clearly differs from the DWT gap.

Figure 14. DMRG estimates of the gap (red crosses), exact thermodynamic gap

(black curve) and DWT gap (green curve), all as a function of β at α = 0.4 (for which

βc ≈ 0.53).

comparison, de Gier and Essler predict a constant value for the gap of−0.00121 . . . above

the dynamical transition, while domain wall theory predicts −0.00102 . . .. The DMRG

measurement is then clearly consistent with the de Gier–Essler prediction, whilst the

difference from the domain wall theory value is significant. Taking this result together

with the Monte Carlo data we conclude that a dynamical transition does indeed occur

at the dGE line, and not where DWT predicts.

5. Discussion of the distinction between static and dynamic subphases

In this paper we have presented numerical evidence, the most convincing of which came

from DMRG calculations, that a dynamical transition occurs in the TASEP at the

dGE line rather than at the subphase boundary. However, this finding in turn raises

Dynamical Transition in the TASEP 18

a number of open questions. For example, does the dGE line—and in particular its

departure from the static subphase boundary—have any physical significance? How is

the static subphase boundary manifested in the eigenvalue spectrum? We discuss the

distinction between the two phase diagrams first in general theoretical terms, and then

with reference to different treatments of the TASEP.

5.1. General theory of nonequilibrium phase transitions

One way to gain a general understanding of nonequilibrium phase transitions is through

the analogue of partition function for a system of size L, ZL [31,32]. It has been shown

that for any system governed by Markovian dynamics, it may be written as

ZL =∏

j 6=0

(−λj) (19)

where (−λj) are the eigenvalues of the Markov transition matrix which defines the

dynamics [32,42]. For a finite, irreducible configuration space there is a unique stationary

state corresponding to the largest eigenvalue λ0 = 0. By the Perron-Frobenious

theorem [43], λ0 is non-degenerate, therefore the gap is given by the second largest

eigenvalue λ1 and the largest relaxation time is τ1 = −1/λ1.

From our knowledge of equilibrium phase transitions, we expect a static phase

transition to occur when limL→∞1LlnZL exhibits nonanalyticities. We see from (19)

that to obtain a static phase transition we require a major restructuring of the eigenvalue

spectrum. To be general and explicit let us consider the nonequilibrium analogue of the

free energy density,

hL =lnZL

L=

1

L

∑

λi 6=0

ln(−λi) . (20)

Consider an arbitrary eigenvalue λi in (20), possessing a nonanalyticity at a critical value

of some parameter. Unless a significant number of other eigenvalues in the spectrum

converge onto the same nonanalyticity in the thermodynamic limit, the effect of the λinonanalyticity is lost as L→ ∞ and (20) and hence hL remain analytic.

On the other hand a dynamical phase transition, as defined above, is only concerned

with the eigenvalue λ1. Therefore a dynamical phase transition does not necessarily

coincide with a static phase transition. One very simple example would be if the two

leading subdominant eigenvalues λ1 and λ2 crossed at some values of a control parameter.

Then the gap would behave in a nonanalytic way but Z would be analytic.

Let us consider more explicitly the TASEP. From the exact solution of the TASEP

[4, 15] the expression for ZL, as defined by (19), is

ZL =(αβ)L+1

β − α[ΦL(α)− ΦL(β)] (21)

where

ΦL(x) =L∑

p=1

p(2L− p− 1)!

L!(L− p)!x−(p+1) (22)

Dynamical Transition in the TASEP 19

and for L large

ΦL(x) ≃

x(1− 2x)

[x(1− x)]L+1x < 1

2

4L√πL

x = 12

x 4L√π(2x− 1)2L3/2

x > 12

. (23)

Now consider, for example, the case α < 1/2. As we increase β a first-order static

phase transition occurs on the line β = α where the dominant contribution to (21)

changes from ΦL(β) to ΦL(α). Thus at the phase transition hL defined in (20) is non-

analytic. The subphase boundary is at β = 1/2 where the asymptotic behaviour of

the subdominant contribution, ΦL(β) changes. As this is a subdominant contribution

to lnZL, hL remains analytic at the sub-phase boundary. At the dGE line there is no

noticeable change in the asymptotic behaviour of ZL and hL is analytic.

5.2. Static vs dynamic transitions within various treatments of the TASEP

We illustrate further the distinctions between the static and dynamics phase diagrams

by collecting exact results for the TASEP together with the predictions of various level

of approximations such as mean field theory and domain wall theory.

Burgers Equation (Mean Field theory) We begin by considering a mean field

description of the system that is given by the Burgers equation,

∂tρ(x, t) = −(1− 2ρ)∂xρ+1

2∂2xρ , (24)

where ρ(x, t) is a density field. Although the Burgers equation has been studied

extensively in the literature, we are not aware of any treatment of the case with

prescribed boundary reservoir densities. In the appendix we give the solution of (24)

subject to the boundary conditions ρ(0) = α and ρ(L) = 1 − β, and arbitrary initial

condition.

The results for the Burgers gap εB may be summarised as follows (restricting

ourselves to the case α < 1/2):

For β < 1/2 εB = − 2|α(1− α)− β(1− β)| (25)

For β > 1/2 εB = − (1− 2α)2

2(26)

The behaviour of the gap is plotted in Fig. 15. Note that a dynamical transition occurs

at β = 1/2. Interestingly, it turns out that there is no change in the density profile at

this value so, in fact, the mean field theory predicts a dynamical transition instead of a

subphase boundary at β = 1/2. Thus even at the level of mean-field theory, the TASEP

exhibits the phenomenon of a dynamic transition that is not accompanied by a static

transition.

Dynamical Transition in the TASEP 20

Figure 15. Comparison of the gap for L → ∞ from the spectrum of the viscous

Burgers equation (24) to the exact thermodynamic gap, for α = 0.1 and βc ≈ 0.675.

Domain wall theory The predictions of domain wall theory were reviewed in

Section 2.4. As we pointed out there, DWT gives a subphase boundary at β = 1/2

along with a dynamical transition: the distinction between the two phase diagrams is

lost. However, in the subphases LDI and HDI, the DWT gives the correct dynamical

gap to order 1/L2.

Exact results As was discussed in Section 2, the exact results for the stationary state

[15,16] and the spectrum [20,21] reveal behaviour that is different from both mean-field

and domain wall theory. In common with the latter, there is a static subphase boundary

at β = 12. Like the mean field theory, the location of the dynamic transition occurs at a

distinct location to the static phase boundary. However, this location αc(β) is nontrivial

and is not predicted by either of the simpler theories. Moreover, the behaviour of the gap

at the static transition point β = α differs between the mean-field and exact theories:

in the former, the gap has a cusp, whereas in the latter it varies analytically.

6. Conclusion

In this work we have provided numerical evidence to confirm the existence of the

dynamical transition line predicted by de Gier and Essler. Ultimately the evidence

came from the approximate, but quantitatively reliable, DMRG technique.

To conclude, we now return to the main open question resulting from this work,

namely that of the physical significance of the dynamical transition that takes place

along the de Gier–Essler line. We remark again that, as was seen in Section 3,

direct measurement of the gap in Monte Carlo simulations was very difficult. The

nonanalyticity in the leading eigenvalue was barely detectable in the stochastic

simulation data. We were therefore unable to identify, for example, whether the system

relaxed to stationarity in a fundamentally different way either side of the transition line.

Dynamical Transition in the TASEP 21

Given the success of the domain wall theory in predicting the gap in part of the

phase diagram, it is tempting to believe that with some refinement it may be able to

explain the dynamical transition and furthermore reveal the physics associated with it.

For example, it has been noted in [21], that DWT may be modified to give the correct

gap throughout the LD1′ region by simply imposing that the right domain density be

ρ+ = 1 − βc in this region. However DWT then no longer predicts a static subphase

boundary at β = 1/2 and also the transition line αc(β) is not predicted.

It may be that some knowledge of the form of the eigenvectors associated to the

low lying states will allow an understanding of what occurs at the dynamical transition.

It would be of interest to extend the DMRG studies to investigate the form of the

eigenfunctions. Also recent analytic work by Simon [44], in which eigenvectors are

constructed for the partially asymmetric exclusion process through the co-ordinate Bethe

ansatz, may help to shed light on the matter.

Acknowledgments

R.A.B. thanks the RCUK for the support of an Academic Fellowship. AP would like to

thank Fabian Essler for hospitality and discussion, and acknowledges financial support

from the EPSRC.

Appendix A. DMRG algorithm for TASEP

In this appendix, we provide a step-by-step description of the DMRG algorithm used

to obtain the results presented in Section 4.

In what follows we make use of three elementary transition matrices for the TASEP.

With the ordering ( , • ) of single-site configurations we define

hl =

(−α 0

α 0

)and hr =

(0 β

0 −β

)(A.1)

for the entry and exit of particles at the left and right boundary sites. In the bulk, and

with the ordering ( , • , • , • • ) of two-site configurations, we define

hb =

0 0 0 0

0 0 1 0

0 0 −1 0

0 0 0 0

. (A.2)

The DMRG algorithm now proceeds as follows.

0. Initialise block transition matrices We partition the system into two blocks, each

of size ℓ. The transition matrix for the left block can be written as

M(ℓ)L = hl ⊗ I⊗ℓ−1 +

ℓ−1∑

k=1

I⊗k−1 ⊗ hb ⊗ I⊗ℓ−k−1 , (A.3)

Dynamical Transition in the TASEP 22

where I is the 2× 2 identity matrix. Likewise, for the right block one has

M(ℓ)R =

ℓ−1∑

k=1

I⊗k−1 ⊗ hb ⊗ I⊗l−k−1 + I⊗ℓ−1 ⊗ hr . (A.4)

1. Diagonalise the transition matrix for the entire system The transition matrix for

the entire system of size 2ℓ, M (2ℓ), is given by

M (2ℓ) =M(ℓ)L ⊗ I⊗ℓ + I⊗ℓ−1 ⊗ hb ⊗ Iℓ−1 + Iℓ ⊗M

(ℓ)R . (A.5)

Now we diagonalise M (2ℓ), first using the Arnoldi algorithm [45] (as implemented

in the ARPACK package [46] and accessed using Mathematica) to find the left

and right ground state eigenvectors 〈φ0|, |ψ0〉. We also want to compute the gap

and associated eigenvectors |ψ1〉 and 〈φ1|. Following [26], we find this can be done

more accurately by ‘shifting’ the transition matrix throughM ′ =M (2ℓ)+∆|ψ0〉〈φ0|sufficiently far that the gap is now the largest eigenvector of M ′.

2. Form reduced density matrices The reduced density matrices are defined as

ρL =1

4TrR(|ψ0〉〈ψ0|+ |φ0〉〈φ0|+ |ψ1〉〈ψ1|+ |φ1〉〈φ1|) (A.6)

ρR =1

4TrL(|ψ0〉〈ψ0|+ |φ0〉〈φ0|+ |ψ1〉〈ψ1|+ |φ1〉〈φ1|) . (A.7)

Here TrL and TrR indicate a trace over the degrees of freedom in the left and right

blocks respectively.

To be clear, let {|i〉L} and {|j〉R} be basis sets for the left and right blocks

respectively. A state vector |ψ〉 across both blocks can then be expanded as

|ψ〉 =∑

ij

cij |i〉L|j〉R . (A.8)

The trace operation TrL(|ψ〉〈ψ|) is then defined as

TrL(|ψ〉〈ψ|) =∑

jj′

(∑

i

cijcij′

)|j〉R〈j′|R . (A.9)

Note that this is an operator acting only on the right block. The corresponding

operation TrR is defined analogously.

3. Diagonalise the density matrices and form a truncated basis set We first find all

eigenvalues and eigenvectors of the symmetric matrices ρL and ρR using a dense

matrix diagonalisation routine from the LAPACK package [47], again accessed using

Mathematica. We then form the matrices OL and OR that have as their columns

the m eigenvectors with largest eigenvalues of ρL and ρR respectively. Then we

construct projectors

PL = OL ⊗ I and PR = I ⊗ OR (A.10)

that will be used in the renormalisation step below.

4. Enlarge system The left block is enlarged by adding a site at its right end. The

transition matrix for this enlarged block is

M(ℓ+1)L =M

(ℓ)L ⊗ I⊗2 + I⊗ℓ−1 ⊗ hb . (A.11)

Dynamical Transition in the TASEP 23

Likewise, the right block is enlarged by adding a site at its left:

M(ℓ+1)R = hb ⊗ Iℓ−1 + I⊗2 ⊗M

(ℓ)R . (A.12)

5. Renormalise the transition matrices for the enlarged system The renormalisation

is performed by projection the enlarged transition matrices onto the subset of

density matrix eigenstates that was retained in step 3. Formally, this is achieved

by computing

M(ℓ+1)L = P T

L M(ℓ+1)L PL and H

(ℓ+1)R = P T

R M(ℓ+1)R PR . (A.13)

Note that after this transformation, the first ℓ sites of the system are no longer

represented in the ‘physical’ basis (i.e., particle-hole configurations). However, the

form of the projectors (A.10) ensures that the last site of the block is represented

in the physical basis, which is essential for the expression (A.5) to be valid at each

stage of the renormalisation.

6. Return to step 1, putting ℓ→ ℓ + 1.

Appendix B. Solution of the Burgers equation with fixed boundary

densities

Here we solve the Burgers equation (i.e., the continuum limit of the TASEP within a

mean-field theory)

∂tρ(x, t) = −(1− 2ρ)∂xρ+1

2∂2xρ . (B.1)

subject to the boundary conditions ρ(0) = α and ρ(L) = 1 − β. We use the Cole-Hopf

transformation [48], which involves the change of variable

ρ(x, t) =1

2

[1 +

∂

∂xln u(x, t)

]. (B.2)

Substituting into (B.1), and integrating with respect to x eventually yields

∂

∂tu(x, t) =

1

2

∂2

∂x2u(x, t) . (B.3)

The boundary conditions on ρ(x, t) transform to the conditions

u′(0, t) = (2α− 1)u(0, t) (B.4)

u′(L, t) = (1− 2β)u(L, t) . (B.5)

We proceed by finding the eigenfunctions φn(x) that satisfy

1

2

d2

dx2φn(x) = λnφn(x) (B.6)

along with the above boundary conditions at x = 0 and x = L. Thus given an initial

condition ρ(x, 0) the function u(x, t) will be given by

u(x, t) =∑

n≥0

cneλntφn(x) (B.7)

Dynamical Transition in the TASEP 24

where the coefficients cn will depend on the initial condition. Inverting the Cole-Hopf

transformation gives

ρ(x, t) =1

2

1 +

φ′0(x)

φ0(x)+

∑n>0 cne

(λn−λ0)tφ0(x)(φn(x)φ0(x)

)′

∑n≥0 cne(λn−λ0)tφn(x)

. (B.8)

The final term, involving the ratio of sums, vanishes as t → ∞, as all λn with n > 0

exceed λ0. Thus the stationary profile is

ρ(x) =1

2

[1 +

φ′0(x)

φ0(x)

]. (B.9)

We can define the Burgers gap, εB via the asymptotic rate of the exponential decay

of the density profile to its stationary form, viz,

εB = limt→∞

d

dtln[ρ(x, t)− ρ(x)] = λ1 − λ0 . (B.10)

The eigenfunctions given by (B.6) themselves take the form

φ(x) = Aeγx +Be−γx (B.11)

whence λ = 12γ2. The allowed values of γ are quantised due to the boundary conditions

(B.4) and (B.5). We first look for solutions with λ > 0, i.e., real γ. Imposing the

boundary conditions leads to the pair of equations

γ(A− B) = (2α− 1)(A+B) (B.12)

γ(AeγL −Be−γL) = (1− 2β)(Aeγx +Be−γx) . (B.13)

These equations are consistent only if

tanh(γL) =2(1− α− β)γ

(1− 2α)(1− 2β) + γ2. (B.14)

In the limit L → ∞, the left-hand side approaches ±1. The resulting equation has a

solution γ = 1− 2α if α < 12and a solution γ = 1− 2β if β < 1

2.

Solutions with a negative λ have purely imaginary γ = ik (k real), and by following

through the same procedure, one finds that the allowed values of k are the roots of

tan(kL) =2(α+ β − 1)k

k2 − (2α− 1)(2β − 1). (B.15)

In the large-L limit, we find that k ∼ 1/L, and hence the associated eigenvalues vanish

as 1/L2 in the thermodynamic limit.

We can now state the L→ ∞ behaviour of the Burgers gap ε as α and β are varied.

When both α and β are smaller than 12, there are two isolated positive eigenvalues

λ = (2α−1)2

2, (2β−1)2

2. In this region, then, the gap behaves as

εB = −|(2α− 1)2 − (2β − 1)2|2

= −2|α(1− α)− β(1− β)| . (B.16)

Like the exact gap (4), this vanishes along the HD-LD coexistence line α = β < 12.

However, unlike the exact gap, the mean-field gap has a nonanalyticity along this line.

Dynamical Transition in the TASEP 25

If α or β is increased above 12, one of the two solutions with positive λ ceases to

exist, and the second-largest eigenvalue λ1 = 0 in the thermodynamic limit. Hence then,

the size of the gap is simply equal to the value of the largest eigenvalue. That is, within

the low-density phase α < 12, β > 1

2,

εB = −(1 − 2α)2

2(B.17)

and within the high-density phase α > 12, β < 1

2

εB = −(1 − 2β)2

2. (B.18)

The Burgers gap thus exhibits nonanalyticities along the coexistence line α = β < 12,

and along the lines α < 12, β = 1

2and α = 1

2, β < 1

2. Thus the dynamic phase diagram

for the Burgers equation appears the same as the static phase diagram, Fig. 1(a), except

that the subphase boundaries have become dynamical transition lines.

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